A081011 a(n) = Fibonacci(4n+3) + 2, or Fibonacci(2n+3)*Lucas(2n).

4, 15, 91, 612, 4183, 28659, 196420, 1346271, 9227467, 63245988, 433494439, 2971215075, 20365011076, 139583862447, 956722026043, 6557470319844, 44945570212855, 308061521170131, 2111485077978052, 14472334024676223

Offset: 0

Author

R. K. Guy, Mar 01 2003

Keywords

• nonn
• easy

Comments

For n>0, a(n) is the area of the trapezoid defined by the four points (F(n+1), F(n+2)), (F(n+2), F(n+1)), (F(n+3), F(n+4)), and (F(n+4), F(n+3)) where F(n) = A000045(n). - J. M. Bergot, May 14 2014

References

• Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Crossrefs

Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers), A187798.

Programs

GAP

List([0..30], n-> Fibonacci(4*n+3) -2); # G. C. Greubel, Jul 14 2019

Magma

[Fibonacci(4*n+3)+2: n in [0..30]]; // Vincenzo Librandi, Apr 18 2011

Maple

with(combinat) for n from 0 to 30 do printf(`%d,`,fibonacci(4*n+3)+2) od # James Sellers, Mar 03 2003

Mathematica

Table[Fibonacci[4n+3] +2, {n,0,30}] (* or *)
Table[Fibonacci[2n+3]*LucasL[2n], {n, 0, 30}] (* Alonso del Arte, Apr 18 2011 *)
LinearRecurrence[{8,-8,1},{4,15,91},30] (* Harvey P. Dale, Apr 22 2017 *)

PARI

vector(30, n, n--; fibonacci(4*n+3)+2) \\ G. C. Greubel, Jul 14 2019

Sage

[fibonacci(4*n+3)+2 for n in (0..30)] # G. C. Greubel, Jul 14 2019

Formula

a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).

G.f.: (4-17*x+3*x^2)/((1-x)*(1-7*x+x^2)). - Colin Barker, Jun 22 2012

Product_{n>=0} (1 - 1/a(n)) = (3-phi)/2 = A187798. - Amiram Eldar, Nov 28 2024

Extensions

More terms from James Sellers, Mar 03 2003